\( a_n = ar^{n-1} = 3 \times 2^{5} = 3 \times 32 = 96 \)

["Understanding the Geometric Sequence: ( a_n = ar^{n-1} ) with Detailed Examples", "The expression ( a_n = ar^{n-1} ) defines a geometric sequence — a fundamental concept in mathematics used across algebra, finance, computer science, and physics. This formula describes the (n)-th term of a geometric sequence where:", "- ( a ) is the first term,\n- ( r ) is the common ratio, and\n- ( n ) is the term’s position in the sequence.", "### What is a Geometric Sequence?", "A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio, ( r ). Unlike arithmetic sequences that increase by a fixed amount, geometric sequences grow or shrink exponentially.", "The general term is expressed as:", "[\na_n = ar^{n-1}\n]", "Here:", "- ( a_n ) = the (n)-th term\n- ( a ) = initial term ((a_1))\n- ( r ) = common ratio (( \frac{a_2}{a_1} = \frac{a_3}{a_2} = \cdots ))\n- ( n ) = term position", "---", "### Example: ( a_n = ar^{n-1} = 3 \ imes 2^{5} = 3 \ imes 32 = 96 )", "Let’s break down the calculation step-by-step using:", "- First term: ( a = 3 )\n- Common ratio: ( r = 2 )\n- Term position: ( n = 6 ), because ( n - 1 = 5 )", "Using the formula:", "[\na_6 = 3 \ imes 2^{5} = 3 \ imes 32 = 96\n]", "This confirms that the 6th term of the sequence is 96.", "### Why is This Useful?", "Geometric sequences model rapid growth or decay patterns — such as:", "- Compound interest in finance (money doubling or growing exponentially over time)\n- Population growth where each generation is a fixed multiple of the previous\n- Virus spread simulations where each infected person infects ( r ) others\n- Computer algorithms that process data in exponential stages", "---", "### How to Identify the Term", "Given a geometric sequence defined by ( a ) and ( r ), finding any term only requires:", "1. Identify the first term ( a ) and ratio ( r )\n2. Plug ( n ) into ( a_n = ar^{n-1} )\n3. Simplify using exponent rules", "For example, if ( a_7 ) were requested with ( a = 3 ), ( r = 2 ):", "[\na_7 = 3 \ imes 2^{7-1} = 3 \ imes 2^6 = 3 \ imes 64 = 192\n]", "---", "### Summary", "The formula ( a_n = ar^{n-1} ) is a powerful tool for predicting values in exponentially increasing sequences. Using actual values like ( a = 3 ) and ( r = 2 ), we derived the 6th term as 96 through basic computation and exponentiation.", "Understanding geometric sequences enhances mathematical fluency and enables modeling of real-world exponential phenomena — from finance to biology to technology.", "---", "Keywords: ( a_n = ar^{n-1} ), geometric sequence, exponential growth, exponential decay, math formula, compound interest, sequence term, common ratio, algebra tutorial, exponential functions", "Meta Description: Explore how ( a_n = 3 \ imes 2^{5} = 96 ) fits in geometric sequences and learn step-by-step how to calculate terms with ( a = 3 ), ( r = 2 ). Ideal for students and math enthusiasts."]